Model 1: In this first example, we take \(m_{\eta } = 20\)  GeV, with the parameters as shown in the caption of Figures 2 and 3 chosen to fit the DM svalue for \(\Omega h^{2}\approx 0.119\)  and the value for \(\langle \sigma _{ann} v\rangle \simeq 1.8 \times 10^{-26}\)  cm\(^{3}\) /s. The lifetime of the light mediator, \(Z^{\prime }\) , must be less than one second in order to guarantee that the \(Z^{\prime }\)  decays before the epoch of big bang nucleosynthesis [1]}. We have chosen \(M_V = 0.5\)  GeV, which implies that \(M_{Z^{\prime }} \simeq 353\)  MeV. Given this \(M_{Z^{\prime }}\)  mass and all the couplings of \(Z^{\prime }\)  with fermions (see appendix), we then find the \(Z^{\prime }\)  decay width, \(\Gamma _{Z^{\prime }} \simeq 1.76\times 10^{-19}\)  GeV. The branching ratios for the \(Z^{\prime }\)  decay channels are found to be 33.9% into \(u\overline{u}\) , 25.4% into \(e^{+}e^{-}\) , 24% in \(\mu ^{+}\mu ^{-}\) , 8.48% into \(d\overline{d}\)  and 8.18% into \(s\overline{s}\) .
